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Solve.
−2/5x−9<9/10
To solve the inequality, we need to isolate the variable x.
Step 1: Let's first simplify the expression on the left-hand side (LHS) of the inequality.
Since -2/5x is the same as (-2/5) times x, we can rewrite the inequality as:
(-2/5) x - 9 < 9/10
Step 2: Next, we need to get rid of the -9 on the LHS. To do this, we can add 9 to both sides of the inequality.
(-2/5) x - 9 + 9 < 9/10 + 9
(-2/5) x < 9/10 + 9
Simplifying further:
(-2/5) x < 9/10 + 90/10
(-2/5) x < 99/10
Step 3: Now, we need to get rid of the coefficient -2/5 in front of x. To do this, we can multiply both sides of the inequality by the reciprocal of -2/5, which is -5/2. However, since we are multiplying by a negative number, we need to flip the inequality symbol.
(-2/5) x * (-5/2) > (99/10) * (-5/2)
x > -(99/10) * (5/2)
After simplifying:
x > -99/4
So, the solution to the inequality is x > -99/4.
why do we need to multiply it by the reciprocal in step 3?
In step 3, we multiply both sides of the inequality by the reciprocal of -2/5 in order to isolate the variable x. This is a common technique used to eliminate a coefficient in front of the variable.
When we multiply both sides of an inequality by a positive number, the direction of the inequality remains the same. For example, if we multiply both sides of x < 5 by 2, we get 2x < 10, and the direction of the inequality remains the same.
However, when we multiply both sides of an inequality by a negative number, the direction of the inequality reverses. For example, if we multiply both sides of x < 5 by -2, we get -2x > -10, and the direction of the inequality changes from less than (<) to greater than (>). This is why we need to flip the inequality symbol.
In step 3, the coefficient in front of x is -2/5, which is a negative number. So, when we multiply both sides of the inequality by its reciprocal (-5/2), we need to flip the inequality symbol to maintain the correct direction.
To solve the inequality −2/5x − 9 < 9/10, we can follow these steps:
Step 1: Simplify the inequality.
Start by adding 9 to both sides of the inequality:
-2/5x - 9 + 9 < 9/10 + 9/10
This simplifies to:
-2/5x < 18/10
Step 2: Simplify the right side.
To simplify further, we can simplify 18/10 to a mixed number or decimal form. Let's keep it in decimal form:
-2/5x < 1.8
Step 3: Divide both sides by -2/5.
Dividing both sides of the inequality by -2/5 requires us to multiply both sides by its reciprocal (invert and multiply):
-2/5x * (5/-2) > 1.8 * (5/-2)
The -2/5's cancel on the left side, and the multiplication on the right side gives us:
x > -9/5
Step 4: Simplify the right side.
The right side, -9/5, can be simplified further. Since -9/5 is already in its simplest form, we can keep it as is. Alternatively, we can also express it as a decimal, which is approximately -1.8.
Therefore, the solution to the inequality −2/5x − 9 < 9/10 is x > -9/5 or x > -1.8.